Transition from geostrophic turbulence to inertia–gravity
waves in the atmospheric energy spectrum
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Citation
Callies, Jorn, Raffaele Ferrari, and Oliver Buhler. “Transition from
Geostrophic Turbulence to Inertia–gravity Waves in the
Atmospheric Energy Spectrum.” Proceedings of the National
Academy of Sciences 111, no. 48 (November 17, 2014):
17033–17038.
As Published
http://dx.doi.org/10.1073/pnas.1410772111
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National Academy of Sciences (U.S.)
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Final published version
Accessed
Thu May 26 22:48:20 EDT 2016
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http://hdl.handle.net/1721.1/97429
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Detailed Terms
Transition from geostrophic turbulence to
inertia–gravity waves in the atmospheric
energy spectrum
Jörn Calliesa,1, Raffaele Ferrarib, and Oliver Bühlerc
a
Massachusetts Institute of Technology and Woods Hole Oceanographic Institution Joint Program in Oceanography and bEarth, Atmospheric, and Planetary
Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139; and cCourant Institute of Mathematical Sciences, New York University, New York,
NY 10012
Midlatitude fluctuations of the atmospheric winds on scales of
thousands of kilometers, the most energetic of such fluctuations,
are strongly constrained by the Earth’s rotation and the atmosphere’s stratification. As a result of these constraints, the flow is
quasi-2D and energy is trapped at large scales—nonlinear turbulent
interactions transfer energy to larger scales, but not to smaller scales.
Aircraft observations of wind and temperature near the tropopause
indicate that fluctuations at horizontal scales smaller than about
500 km are more energetic than expected from these quasi-2D
dynamics. We present an analysis of the observations that indicates
that these smaller-scale motions are due to approximately linear
inertia–gravity waves, contrary to recent claims that these scales
are strongly turbulent. Specifically, the aircraft velocity and temperature measurements are separated into two components: one
due to the quasi-2D dynamics and one due to linear inertia–gravity
waves. Quasi-2D dynamics dominate at scales larger than 500 km;
inertia–gravity waves dominate at scales smaller than 500 km.
|
meteorology atmospheric dynamics
inertia–gravity waves
| geostrophic turbulence |
T
he midlatitude high- and low-pressure systems visible in
weather maps are associated with winds and temperature
fluctuations that we experience as weather. These fluctuations
arise from a baroclinic instability of the mean zonal winds at
horizontal scales of a few thousand kilometers, commonly referred to as the synoptic scales (1–3). The combined effects of
rotation and stratification constrain the synoptic-scale winds to
be nearly horizontal and to satisfy geostrophic balance, a balance
between the force exerted by the changes in pressure and the
Coriolis force resulting from Earth’s rotation. It is an open
question whether the same constraints dominate in the mesoscale range (i.e., at scales of 10–500 km), or whether qualitatively
different dynamics govern flows at these scales.
The synoptic-scale flows are turbulent in the sense that nonlinear scale interactions, which lie at the core of the difficulty to
predict the weather, exchange energy between different scales of
motion (4–7). Under the constraints of rotation and stratification,
the synoptic-scale winds are approximately 2D and nondivergent
(8, 9). In 2D flows, nonlinear scale interactions tend to transfer
energy to larger scales, that is, the synoptic-scale pressure anomalies often merge and form larger ones, contrary to nonlinear scale
interactions in 3D flows, which tend to transfer energy to smaller
scales (10). Little energy is thus transferred to scales smaller than
those at which the synoptic-scale fluctuations are generated
through instabilities. Theory and numerical simulations predict
that the energy per unit horizontal wavenumber k decays as rapidly
as k−3 at wavenumbers larger than the wavenumber corresponding
to the instability scale (9, 11). This predicted kinetic energy spectrum is roughly consistent with synoptic-scale observations (9, 12).
Long-range passenger aircraft have been instrumented to
collect velocity and temperature measurements as part of the
Global Atmospheric Sampling Program in the 1970s and the
Measurement of Ozone and Water Vapor by Airbus In-Service
www.pnas.org/cgi/doi/10.1073/pnas.1410772111
Aircraft (MOZAIC) project in the 1990s and 2000s. The
resulting dataset, described in Materials and Methods, consists of
tens of thousands of flights. Because aircraft travel at altitudes
between 9 and 14 km, the data largely reflect the upper troposphere and lower stratosphere, near the tropopause. These
measurements confirm that the kinetic energy spectrum drops as
k−3 in the synoptic wavenumber range, but there is a transition in
behavior at a scale of about 500 km (13) (Fig. 1A). In the mesoscale range, at scales smaller than 500 km, the kinetic energy
spectrum decays more slowly, roughly like k−5=3 (13–15).
The measured kinetic energy spectrum is intriguing, because it
agrees so well with Charney’s theory of geostrophic turbulence at
the synoptic scales (9) but deviates from that prediction at the
mesoscale. The transition to the flatter k−5=3 mesoscale spectrum
has been interpreted as the signature of small-scale geostrophic
flows generated by convective events (14, 16, 17), as the development of fronts at the edge of synoptic-scale cyclones and
anticyclones at the top of the troposphere (equivalent to the
warm and cold mesoscale fronts we experience at the Earth’s
surface) (18), or as the signature of stratified turbulence at scales
where the rotational constraints become less important (19).
These explanations of the synoptic-to-mesoscale transition invoke turbulent dynamics and strong interactions between the
synoptic and mesoscale flows.
A rotating and stratified atmosphere, however, supports an
additional, much faster set of motions: inertia–gravity waves.
These are internal gravity waves, modified by the effect of rotation, that have periods of several minutes to a few hours. In
contrast to the strongly nonlinear, turbulent synoptic-scale flow,
these motions are wave-like and at small amplitude they are
approximately governed by linear dynamics (20). It has been
Significance
High- and low-pressure systems, commonly referred to as
synoptic systems, are the most energetic fluctuations of wind
and temperature in the midlatitude troposphere. Synoptic
systems are a few thousand kilometers in scale and are governed by a balance between the pressure gradient force and
the Coriolis force. Observations collected near the tropopause
by commercial aircraft indicate a change in dynamics at horizontal scales smaller than about 500 km. Smaller-scale fluctuations are shown to be dominated by inertia–gravity waves,
waves that propagate on vertical density gradients but are
influenced by Earth’s rotation.
Author contributions: J.C. designed research; J.C., R.F., and O.B. performed research; J.C.
analyzed data; and J.C., R.F., and O.B. wrote the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
Freely available online through the PNAS open access option.
1
To whom correspondence should be addressed. Email: joernc@mit.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1410772111/-/DCSupplemental.
PNAS | December 2, 2014 | vol. 111 | no. 48 | 17033–17038
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Edited* by Kerry A. Emanuel, Massachusetts Institute of Technology, Cambridge, MA, and approved October 17, 2014 (received for review June 14, 2014)
A
B
C
Fig. 1. Observed wavenumber spectra of near-tropopause midlatitude winds and decomposition into geostrophic component and inertia–gravity wave
component. (A) Observed spectra of longitudinal kinetic energy Su ðkÞ, transverse kinetic energy Sv ðkÞ, and potential energy Sb ðkÞ. (B) Helmholtz decomposition of the observed kinetic energy spectrum KðkÞ into its rotational and divergent components K ψ ðkÞ and K ϕ ðkÞ. (C) Partitioning of the total energy
spectrum EðkÞ into the diagnosed inertia–gravity wave component Ew ðkÞ and the residual geostrophic component Eg ðkÞ.
proposed that the mesoscale energy is dominated by inertia–
gravity waves (21, 22), which are easily excited by any fast fluctuation of the atmospheric flows (23). In this explanation of the
mesoscale part of the spectrum, linear inertia–gravity waves and
nonlinear synoptic-scale turbulence coexist with little interaction.
In this paper, we present an analysis of the MOZAIC data that
uses a decomposition method recently developed by Bühler et al.
(24). This analysis provides compelling evidence that linear inertia–
gravity waves indeed dominate the observations in the mesoscale.
Theories for the Synoptic-to-Mesoscale Transition
Dewan (21) first suggested that the mesoscale energy is dominated by a continuum of weakly nonlinear inertia–gravity waves.
VanZandt (22) showed that the mesoscale spectra of horizontal
wind fluctuations as a function of horizontal wavenumber, vertical wavenumber, and frequency are related through the dispersion and polarization relations of inertia–gravity waves. The
dominance of inertia–gravity waves at the mesoscale, however,
seemed inconsistent with vertical velocity frequency spectra
measured from radar in light-wind conditions (25). More recently, Vincent and Eckerman (26) showed that the signature of
inertia–gravity waves is recovered after correcting for Dopplershift effects in radar observations.
In the last two decades, the interpretation of the synoptic-tomesoscale transition in terms of inertia–gravity wave dynamics
has received little attention. The k−5=3 power-law dependence of
the mesoscale spectrum has instead been interpreted as evidence
that the mesoscale is strongly turbulent. The strong nonlinear
interactions characteristic of turbulent flows continuously redistribute energy across wavenumbers and are known to result in
power-law energy distributions (4, 7). A number of competing
turbulent theories have been proposed.
The earliest turbulent theory argues that the mesoscale spectrum is due to an inverse cascade of energy injected at even
smaller scales, for example by convective activity (14, 16, 17).
The hypothesis is that nonlinear mesoscale interactions, much
like at synoptic scales, transfer energy to larger scales because of
the constraints of rotation and stratification. The kinetic energy
spectrum in such a quasi-2D inverse cascade scales like k−5=3 at
wavenumbers smaller than the injection scale (11), in contrast to
Charney’s k−3 spectrum, which develops at wavenumbers larger
than the injection scale. The theory does not predict at which scale
the dynamics switch from the synoptic to the mesoscale regime.
The second turbulent theory proposes that the flat mesoscale
spectrum is the signature of sharp temperature fronts, which
17034 | www.pnas.org/cgi/doi/10.1073/pnas.1410772111
develop when synoptic-scale flows intersect a rigid boundary, like
the Earth’s surface, or a strongly stratified layer, like the tropopause (18). Importantly, the winds associated with the temperature fronts are still largely in geostrophic balance. In this
view, the k−5=3 mesoscale spectrum is a feature of measurements
taken at the tropopause level, the cruising altitude of long-range
commercial aircraft, but should not appear in the mid troposphere. This prediction cannot be tested with available data.
The third turbulent theory proposes that the k−5=3 mesoscale
spectrum emerges at the scales where the flows escape the rotational constraint and energy can be transferred to smaller
scales (19). Turbulent flows constrained by stratification, but not
rotation, are collectively known as “stratified turbulence.” The
forward energy cascade is achieved by the overturning of layerlike structures. These flows are not in geostrophic balance, and
thus differ from the quasi-2D dynamics of the previous two
theories, and they are strongly nonlinear, and thus differ from
approximately linear inertia–gravity waves.
Inertia–Gravity Waves and Geostrophic Flow
The understanding that atmospheric winds are composed of slow
flows in approximate geostrophic balance and fast inertia–gravity
waves has been the foundation for much progress in atmospheric
science. The first numerical weather predictions were based
on quasi-geostrophic dynamics, an approximation to the more
complete primitive equations that filters out inertia–gravity
waves (27–29). In the troposphere and lower stratosphere,
inertia–gravity waves typically have small amplitudes and therefore interact only weakly with geostrophic flows. Although there
is a growing appreciation of rare instances of inertia–gravity
waves directly influencing sensitive weather patterns (30), strong
interactions between inertia–gravity waves and the geostrophic
flow are typically confined to the middle and upper atmosphere,
where the wave amplitude becomes large enough to allow for
breaking of inertia–gravity waves and the concomitant drag force
on the geostrophic flow that is well known to be crucial for the
global angular momentum budget of the atmosphere (23).
In the deep ocean, breaking inertia–gravity waves mix heat and
carbon. This leading-order effect has led to intensive study of the
oceanic inertia–gravity wave field. It is composed of a continuous
spectrum of linear waves together with isolated peaks at the inertial
and tidal frequencies (31). Similar to the lower atmosphere, these
linear waves interact only weakly with the geostrophic flow. Only at
small vertical scales of a few tens of meters do the waves break.
Callies et al.
Decomposition
From the MOZAIC aircraft observations of wind near the
midlatitude troposphere, we compute power spectra of the longitudinal (along-track) velocity u and transverse (across-track)
uðkÞj2 i and Sv ðkÞ = hj^vðkÞj2 i, where the caret
velocity v, Su ðkÞ = hj^
denotes a Fourier transform and the angle brackets an average over flights. From the temperature observations, we compute the potential energy spectrum Sb ðkÞ = hj^bðkÞj2 i=N 2 , where
b = gðθ − θ0 Þ=θ0 is buoyancy, g = 9.81 ms−2 is the gravitational
acceleration, θ0 = 340 K is the reference potential temperature,
and N2 is the average vertical gradient of b. Potential temperature
θ is the temperature of an air parcel corrected for dynamically
irrelevant compression effects. We use a typical stratification of
the lower stratosphere, N = 0.02 s−1, estimated from the ERAInterim reanalysis (33). Fig. 1A shows that these MOZAIC spectra
display the transition from a steep synoptic range to a flat mesoscale range at about 500 km.
If simultaneous wind and temperature observations were
available in space and time, one could directly test whether the
dispersion and polarization relations of inertia–gravity waves are
satisfied by mesoscale motions. One could further separate out
inertia–gravity waves and geostrophic flows, because inertia–
gravity waves are restricted to frequencies between the Coriolis
frequency f (equal to twice the rotation rate of the Earth multiplied by the sine of the latitude) and the buoyancy frequency N
(the frequency at which a vertically displaced parcel of air will
oscillate within the stably stratified atmosphere), whereas geostrophic flows evolve on much longer time scales. However, it is
extremely difficult to collect simultaneous measurements of mesoscale fluctuations of winds and temperature in space and time.
Bühler et al. (24) have recently shown that the decomposition
can be achieved from space-only measurements, provided that
concurrent observations of horizontal velocities and temperature
are available. Applying this new decomposition to the MOZAIC
data produces two powerful arguments in support of the hypothesis
that the mesoscale spectrum is dominated by inertia–gravity waves.
First, assuming that the flow is an uncorrelated superposition
of a geostrophic flow and inertia–gravity waves, we diagnose the
inertia–gravity wave component of the total energy, the sum of
the kinetic and potential energies, solely based on the observed
horizontal velocities. We then show that the thus predicted
inertia–gravity wave energy spectrum closely matches the observed total energy spectrum in the mesoscale range. This indicates that the mesoscale potential energy spectrum, predicted by
this procedure, is consistent with linear wave theory.
Second, assuming that the geostrophic component of the flow
obeys Charney’s isotropy relation for geostrophic turbulence (9),
we decompose the three observed individual spectra of longitudinal kinetic energy, transverse kinetic energy, and potential
energy into a geostrophic component and an inertia–gravity wave
component. The diagnosed inertia–gravity wave spectra closely
match the observed spectra in the mesoscale range. This is another powerful test of the mesoscale flow’s consistency with the
dispersion and polarization relations of inertia–gravity waves.
Helmholtz Decomposition
Fig. 1A shows that in the synoptic range the spectra approximately satisfy Charney’s prediction for geostrophic turbulence:
Su ðkÞ = Sb ðkÞ and Sv ðkÞ = 3Su ðkÞ (9, 12). At the transition to the
mesoscale, all three spectra converge. As shown by Lindborg (34,
Callies et al.
35) and in the following, this convergence is evidence that the
flow is no more in geostrophic balance at leading order.
Any horizontal flow field can be decomposed into its rotational and divergent components: u = −ψ y + ϕx , v = ψ x + ϕy ,
where ψ is the streamfunction, ϕ is the velocity potential, x is the
along-track coordinate, and y is the across-track coordinate. If
the flow is statistically isotropic horizontally and ψ and ϕ are
uncorrelated (as is the case for a superposition of geostrophic
flow and linear inertia–gravity waves), the spectra Su ðkÞ and
Sv ðkÞ can be written in terms of spectral functions associated with
ψ and ϕ (Materials and Methods):
d ϕ
D ðkÞ;
dk
[1]
d ψ
D ðkÞ + Dϕ ðkÞ:
dk
[2]
Su ðkÞ = Dψ ðkÞ − k
Sv ðkÞ = −k
The spectral functions Dψ ðkÞ and Dϕ ðkÞ can easily be computed
from the observed Su ðkÞ and Sv ðkÞ by solving the system of ordinary differential Eqs. 1 and 2 (Materials and Methods). Using
Dψ ðkÞ and Dϕ ðkÞ, the observed kinetic energy spectrum
KðkÞ = 12 ½Su ðkÞ + Sv ðkÞ can be decomposed into its rotational
and divergent components,
1
d
1−k
Dψ ðkÞ;
K ψ ðkÞ =
[3]
2
dk
K ϕ ðkÞ =
1
d
1−k
Dϕ ðkÞ:
2
dk
[4]
The Helmholtz decomposition of the MOZAIC kinetic energy
spectrum in Fig. 1B shows that the rotational component dominates in the synoptic range, whereas the divergent component
becomes of the same order at the transition to the mesoscale
range. In the mesoscale range, the divergent component slightly
dominates over the rotational component.
The dominance of the rotational component in the synoptic
scales is consistent with Charney’s geostrophic turbulence, because quasi-geostrophic flow is to leading order horizontally
nondivergent. The significant divergent component in the mesoscale, however, is inconsistent with the mesoscale theories that
rely on a leading-order geostrophic balance, namely the inversecascade theory and the frontogenesis theory. Instead, it points to
the dominance of ageostrophic dynamics.
Lindborg (35) also found that the rotational and divergent
components of the flow are of the same order in the mesoscale
range. In his analysis, based on curve fitting and selective Fourier
transforming, the rotational component slightly dominated the
divergent component in the mesoscale range. He argued that this
was inconsistent with inertia–gravity waves, for which he expected
the divergent component to be much larger than the rotational
component. If the inertia–gravity wave field is dominated by
near-inertial waves, however, as suggested by balloon measurements in the lower stratosphere (36), the rotational component is expected to be of the same order as the divergent
component. In the following section, we show that the mesoscale
signal is indeed consistent with linear inertia–gravity wave dynamics (i.e., with the dispersion and polarization relations of
hydrostatic inertia–gravity waves).
Decomposition of the Total Energy Spectrum into Geostrophic and
Inertia–Gravity Wave Components. Geostrophic flows are hori-
zontally nondivergent and therefore only have a rotational
component, whereas inertia–gravity waves have both a rotational
and a divergent component. To perform the decomposition into
these two classes of motion, we note that the component of the
total energy spectrum EðkÞ = 12 ½Su ðkÞ + Sv ðkÞ + Sb ðkÞ that is due
to hydrostatic inertia–gravity waves can be diagnosed from Dϕ ðkÞ
alone (Materials and Methods):
PNAS | December 2, 2014 | vol. 111 | no. 48 | 17035
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Callies and coworkers (24, 32) have recently shown that the
energy spectra of oceanic flows are dominated by geostrophic
flows at large scales and by inertia–gravity waves at small scales.
The transition between the two classes of motion occurs at scales
of 10–100 km, depending on the relative strength of the geostrophic eddies and the waves. In what follows, we show that the
synoptic-to-mesoscale transition in the atmospheric energy spectrum is likely an equivalent transition from geostrophic to inertia–
gravity wave dynamics.
d
Ew ðkÞ = 2K ðkÞ = 1 − k
Dϕ ðkÞ;
dk
ϕ
[5]
where the subscript w designates the inertia–gravity wave component. This somewhat surprising result follows directly from linear
inertia–gravity wave dynamics, if horizontal isotropy and vertical
homogeneity are assumed. Provided there is no additional type of
motion, the residual of the observed total energy spectrum can be
attributed to a geostrophic flow, Eg ðkÞ = EðkÞ − Ew ðkÞ, where the
subscript g designates the geostrophic component.
The decomposition into geostrophic flow and inertia–gravity
waves of the MOZAIC data are shown in Fig. 1C. The mesoscale
range is dominated by inertia–gravity waves, which do not contribute much energy at the synoptic scales. The residual spectrum
(i.e., the total spectrum minus the inertia–gravity wave component)
dominates at the synoptic scales. This component can be confidently attributed to geostrophic flows, because we have shown that
the spectrum in the synoptic range is purely rotational and hence
horizontally nondivergent. At the transition scale, the geostrophic
component of the total energy keeps falling off steeply—the transition seems to be due to inertia–gravity waves becoming dominant
in the mesoscale range. This is the main result of this paper.
Notice that the geostrophic spectrum keeps falling off steeply
past the transition at 500 km but eventually flattens out at smaller
scales. This flattening is likely an artifact, because at these scales
the geostrophic component makes up a small fraction of the observed signal. It is quite possible that the flattening is due to noise
or biases introduced by the interpolation procedure or by truncation errors in the reported wind and temperature data.
Decomposition of Kinetic and Potential Energy Spectra. We have
shown that the total energy spectrum can be decomposed into its
geostrophic and inertia–gravity wave components. To confirm
that the observed spectra are consistent with geostrophic dynamics at synoptic scales and with inertia–gravity wave dynamics
at mesoscales, we now decompose into its two components each
of the atmospheric spectra, the longitudinal and transverse kinetic energy spectra, Su ðkÞ and Sv ðkÞ, and the potential energy
spectrum, Sb ðkÞ. This can be done if one makes one further
assumption. Following Charney and results from numerical
simulations of geostrophic turbulence, the geostrophic streamfunction is assumed to be 3D isotropic, with the vertical coordinate rescaled by f =N (9). This implies Sug ðkÞ = Sbg ðkÞ, a relation
that we noted is satisfied by the observed spectra in the synoptic
range (Fig. 1A).
This decomposition—obtained by applying Eqs. 15–17 and
20–22, given in Materials and Methods—confirms the main conclusion that the observed synoptic-scale flow is consistent with
geostrophic dynamics and that the observed mesoscale flow is
consistent with inertia–gravity wave dynamics. In the synoptic
range, the observed spectra match the diagnosed geostrophic
components (Fig. 2A). At the transition scale, the diagnosed
geostrophic spectra start deviating from the observed spectra and
keep falling off steeply. At this scale, the diagnosed inertia–
gravity wave spectra become comparable to the observed spectra
and start matching them in the mesoscale range (Fig. 2B).
Discussion
Our analysis shows that the aircraft observations are consistent
with a geostrophic flow that dominates the synoptic range and
with inertia–gravity waves that dominate the mesoscale range.
This conclusion is predicated on the assumption that the total
observed flow is a superposition of geostrophic flow and inertia–
gravity waves and that these two components are uncorrelated
and horizontally isotropic.
In accord with Lindborg’s result (34), our analysis conclusively
shows that mesoscale flows are not in geostrophic balance, thus
falsifying previous suggestions that mesoscale spectra represent
geostrophic eddies generated by atmospheric convective events
or geostrophically balanced fronts at the tropopause. Our analysis
17036 | www.pnas.org/cgi/doi/10.1073/pnas.1410772111
then further shows that the observed mesoscale spectra Su ðkÞ,
Sv ðkÞ, and Sb ðkÞ are consistent with the dispersion and polarization relations of linear hydrostatic inertia–gravity waves. Presumably, it would be very surprising if the strongly nonlinear
ageostrophic flows characteristic of stratified turbulence were to
yield the same relations between these mesoscale spectra as the
linear waves.
A similar transition between a dominant geostrophic flow at
large scales and dominant inertia–gravity waves at small scales is
well established in the oceanic spectra (32), providing further
support for our interpretation of the synoptic-to-mesoscale
transition. The ocean, like the atmosphere, is a strongly rotating
and stratified fluid. The most energetic large-scale fluctuations
are generated by baroclinic instabilities in both fluids, whereas
small-scale inertia–gravity waves are triggered by any fast perturbation. A posteriori, it should not be surprising that both
fluids exhibit a transition from geostrophic dynamics at large
scales to inertia–gravity wave dynamics at small scales.
That inertia–gravity waves dominate the mesoscale spectrum
does not mean that the atmosphere is filled with a nearly uniform
and stationary wave field, as seems to be the case in the ocean
(31). The atmospheric wave field is likely highly intermittent in
both space and time (23). Our result merely suggests that, on
average, inertia–gravity waves dominate the mesoscale range.
A good understanding of what sets the shape of the average
atmospheric inertia–gravity wave spectrum is lacking. The oceanic inertia–gravity wave spectrum, however, has been shown to
be an equilibrium solution of weakly interacting inertia–gravity
waves with slopes close to k−5=3 (37, 38). Although the residence
time of waves in the atmosphere is shorter and shears provided
by jet streams stronger, weak turbulence theory may similarly
yield insight into the atmospheric inertia–gravity wave field.
The spatial patterns of mesoscale energy are also consistent
with the dominance of inertia–gravity waves at the mesoscales.
The aircraft spectra in the mesoscale range are up to six times
larger in mountainous regions than over flat terrain (39). The
inertia–gravity wave activity is expected to be enhanced over
mountains, where lee waves are excited by large- and synopticscale flows impinging on topography. Stratified turbulence,
however, is fed directly by synoptic-scale energy and therefore is
not expected to be enhanced over mountainous regions.
High-resolution numerical models reproduce the synoptic-tomesoscale transition (40). Model results are consistent with the
observation that the synoptic range is dominated by rotational flow,
whereas the rotational and divergent components are of the same
order in the mesoscale range (41, 42). Models also show a pattern of
enhanced mesoscale energies in regions of high topography (42).
The simulations further show that the mesoscale spectrum is not the
result of stratified turbulence (43), in agreement with our conclusion that they are the signature of inertia–gravity waves.
The emergent picture is relatively simple. Geostrophic synoptic-scale baroclinic disturbances force a forward enstrophy
cascade that continues through the synoptic-to-mesoscale transition. The k−3 or slightly steeper geostrophic spectrum is
masked by inertia–gravity waves at scales smaller than 500 km.
This picture does not rule out the possibility that some fraction
of the energy in the mesoscale spectra is associated with fronts
and stratified turbulence, but these contributions must be small.
The result of this paper may also have some implications for
the theoretical predictability of atmospheric flows. Lorenz (6)
argued that a turbulent flow with a k−5=3 kinetic energy spectrum has a finite predictability time, which is of the order of the
eddy turnover time. More and more accurate knowledge of the
initial state cannot push forecasts beyond that limit—even if we
had a perfect model. In contrast, turbulent flows with a k−3
kinetic energy spectrum do not have such a predictability limit
and ever more accurate initial conditions can lead to ever
longer forecasts.
If the flat mesoscale spectrum were due to a turbulent cascade,
that would pose a limit on predictability of synoptic systems (7, 44).
Observing and modeling the atmosphere beyond the transition
Callies et al.
A
B
segments are discarded if they are shorter than 6,000 km, the average sample
spacing is coarser than 1.2 km, or the deviation from the great circle is greater
than 2°. The data are then linearly interpolated onto a regular grid with 1-km
spacing. Data at pressures larger than 350 hPa are discarded. Subsequently, for
each flight, data deviating more than 1 km in altitude from the mean altitude
are also discarded. Temperature data are adjusted to account for remaining
variations in flight altitude, assuming a constant stratification N = 0.02 s−1, but
this correction is of no consequence for the results discussed in the main paper.
Nastrom and Gage (13) and Cho and Lindborg (15) showed that the spectral
shapes are qualitatively the same in the upper troposphere and lower stratosphere, and thus we do not separate the data into vertical bins. Spectra are
computed by applying a Hann window, compensating for the variance loss,
performing a discrete Fourier transform, and averaging over all 458 segments.
The windowing is necessary to prevent spectral leakage of synoptic-scale energy
into the mesoscale. Spectra at wavelengths smaller than 20 km are discarded
because they are potentially affected by the interpolation procedure or by
truncation errors in the reported data. Locations were reported in longitude/
latitude with an accuracy of 0.01°, zonal and meridional winds were reported
with an accuracy of 0.01 m s−1, and temperatures were reported with an accuracy of 0.01 K. These noise levels do not affect the spectra on scales larger than
20 km. We also discard the largest resolved wavelength, because the power at
this wavelength is reduced artificially by the window.
Fig. 2. Decomposition of observed wavenumber spectra into the geostrophic and inertia–gravity wave components. (A) Diagnosed geostrophic
component of the spectra of longitudinal kinetic energy Sug ðkÞ, transverse
kinetic energy Svg ðkÞ, and potential energy Sbg ðkÞ (heavy lines) and observed
spectra for reference (faint lines). Note that by construction Sug ðkÞ = Sbg ðkÞ, so
that the red and black heavy lines are on top of each other. (B) Diagnosed
inertia–gravity wave component of the spectra of longitudinal kinetic energy Suw ðkÞ, transverse kinetic energy Svw ðkÞ, and potential energy Sbw ðkÞ
(heavy lines) and observed spectra for reference (faint lines). A blowup of
the mesoscale range of this panel can be found in Fig. S1.
at 500 km would yield rapidly diminishing returns in predictability. If this part of the spectrum is due to inertia–gravity
waves, however, improving observational systems and forecast
models may not prove as futile. Inertia–gravity waves do not
propagate errors in the same way as the turbulent flows discussed by Lorenz. If the geostrophic component of the mesoscale
flow were to be observed, despite the dominance of inertia–
gravity waves, the forecast times of synoptic systems could potentially be extended considerably.
It should be noted, however, that other processes not considered in this theoretical argument may affect predictability.
Moist convective processes, for example, lead to rapid growth of
errors that can leak into the geostrophic flow (45). Currently, the
practical predictability of the weather is likely limited by inadequate representation of such processes.
Materials and Methods
Aircraft Data. The spectra shown in Fig. 1A are calculated from wind and temperature measurements obtained by the MOZAIC program, which equipped
commercial aircraft with instrumentation to measure trace gases, but also
records wind speed and direction from the onboard computer. The data used
here were obtained in 2002–2010 and are restricted to the northern hemisphere
midlatitudes (30–60° latitude). Great circles are fit to the flight paths and
Callies et al.
Su ðk,lÞ = l 2 Sψ ðk,lÞ + k2 Sϕ ðk,lÞ,
[6]
Sv ðk,lÞ = k2 Sψ ðk,lÞ + l 2 Sϕ ðk,lÞ,
[7]
where l is the across-track wavenumber. Integration over l and some manipulation gives Eqs. 1 and 2 with
Dψ ðkÞ =
Dϕ ðkÞ =
1
π
1
π
Z∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kh2 − k2 Sψ ðkh Þdkh ,
[8]
k
Z∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kh2 − k2 Sϕ ðkh Þdkh ,
[9]
k
spectra
of ψ and ϕ, related to
where Sψ ðkh Þ and Sϕ ðkh Þ are the 2D isotropic
ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
the 2D spectra by Sðkh Þ = 2πkh Sðk,lÞ, kh = k2 + l 2 is the magnitude of the
horizontal wavenumber vector, and k ≥ 0 is the along-track wavenumber.
Eqs. 1 and 2 can be solved explicitly, given the boundary conditions
Dψ ð∞Þ = 0 and Dϕ ð∞Þ = 0,
Dψ ðsÞ =
Z∞
½Su ðσÞsinhðs − σÞ + Sv ðσÞcoshðs − σÞdσ,
[10]
½Su ðσÞcoshðs − σÞ + Sv ðσÞsinhðs − σÞdσ,
[11]
s
Dϕ ðsÞ =
Z∞
s
where for convenience the coordinate was transformed to s = ln k.
Relation 5 follows from the dispersion and polarization relations of hydrostatic
inertia–gravity waves. Combining the vorticity and continuity equations of the
linearized primitive equations yields ∇2 ψ t = −f ∇2 ϕ (20), which implies that
Sψ ðk,l,ωÞ =
f2 ϕ
S ðk,l,ωÞ
ω2
[12]
and thus, with the use of Eqs. 6 and 7,
PNAS | December 2, 2014 | vol. 111 | no. 48 | 17037
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Details of Decomposition. We here give more detail on the decomposition
techniques used to analyze the aircraft spectra. Bühler et al. (24) give a more
comprehensive description of these techniques and illustrate their skill to
analyze oceanic spectra.
Let u and v be horizontal velocity components defined in the x–y plane
plane with x aligned with the aircraft track, so u is the longitudinal (alongtrack) component and v is the transverse (across-track) component. The time
t and altitude z are considered fixed during the measurement, so they will
be ignored. A general 2D flow has a Helmholtz decomposition into rotational and divergent components of the form u = −ψ y + ϕx and v = ψ x + ϕy .
The functions ψ and ϕ are uniquely determined in terms of the velocity field
with doubly periodic boundary conditions.
Progress with the statistical theory is possible if ψðx,yÞ and ϕðx,yÞ are
uncorrelated. We can then write the 2D power spectra of u and v as
f2 Su ðk,l,ωÞ + Sv ðk,l,ωÞ = 1 + 2 k2 + l 2 Sϕ ðk,l,ωÞ:
ω
[13]
The linear buoyancy
equation
is bt + N2 w = 0 and the potential energy
2
bðkÞ æ=N2 can also be related to the spectrum of the
spectrum Sb ðkÞ = Æ^
Because Eg ðkÞ = EðkÞ − Ew ðkÞ can be diagnosed from the observations and
Eq. 5, this can be solved for Dψg ðkÞ:
Dψg ðsÞ = 2
Z∞
Eg ðσÞe2ðs−σÞ dσ,
[19]
s
velocity potential,
f2 S ðk,l,ωÞ = 1 − 2 k2 + l 2 Sϕ ðk,l,ωÞ,
ω
b
[14]
if uncorrelated plane waves or equivalently vertical homogeneity is assumed. A slowly varying background is allowed. Adding Eqs. 13 and 14 then
eliminates the dependence on ω, so that Eq. 5 follows by integrating over l
and ω.
The decomposition of the three individual spectra can be achieved by
decomposing Dψ ðkÞ = Dψg ðkÞ + Dψw ðkÞ. The divergent part Dϕ ðkÞ needs no such
decomposition, because the geostrophic component of the flow is
divergence-free. Using
Sug ðkÞ = Dψg ðkÞ,
Svg ðkÞ = −k
d ψ
D ðkÞ,
dk g
[15]
where the boundary condition Dψg ð∞Þ = 0 was used and the coordinate was
again transformed to s = ln k for convenience. The decomposition is now
complete. The wave spectra are
Suw ðkÞ = Dψw ðkÞ − k
Svw ðkÞ = −k
d ϕ
D ðkÞ,
dk w
d ψ
D ðkÞ + Dϕw ðkÞ,
dk w
d ϕ
Dw ðkÞ − Dψw ðkÞ ,
Sbw ðkÞ = 1 − k
dk
[20]
[21]
[22]
where Dψw ðkÞ = Dψ ðkÞ − Dψg ðkÞ, Dϕw ðkÞ = Dϕ ðkÞ, and Eq. 5 was used to deduce
the last equation.
[16]
where Charney’s assumption Sbg ðkÞ = Sug ðkÞ was applied, we can write the
total energy spectrum of the geostrophic component Eg ðkÞ = 12 ½Sug ðkÞ +
Svg ðkÞ + Sbg ðkÞ as
k d
Dψ ðkÞ:
[18]
Eg ðkÞ = 1 −
2 dk g
ACKNOWLEDGMENTS. We thank the principal investigators of the MOZAIC
project, the European Commission for supporting the MOZAIC project,
ETHER [Centre National d’Etudes Spatiales (CNES)–CNRS/Institut National des
Sciences de l’Univers (INSU)] for hosting the database, and the participating
airliners for transporting the instrumentation free of charge. J.C. and R.F.
thank Glenn Flierl for offering useful feedback during the preparation of the
manuscript and acknowledge financial support under Grants ONR-N-0001409-1-0458 and NSF-CMG-1024198. O.B. gratefully acknowledges financial
support under Grants NSF-CMG-1024180, NSF-DMS-1312159, and NSF-DMS1009213.
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Sbg ðkÞ = Dψg ðkÞ,
17038 | www.pnas.org/cgi/doi/10.1073/pnas.1410772111
[17]
Callies et al.